Transcript Chapter 2

Chapter 2
Linear Programming:
Fundamentals
Linear Programming (LP)
• Linear programming is a
optimization model with an
objective and a set of
constraints.
• LP is a model for restricted
decision making.
An Application Example
Resource Requirements
Labor
Clay
Products
(hr/unit)
(lb/unit)
Bowl
1
4
Mug
2
3
Resource
40 hours
120 lb
Available
Profit
($/unit)
40
50
Find how many bowls and mugs should be produced to
maximize the profit.
LP Components
• Decision variables - their values are to
be found in the solution.
• One linear objective function.
• Linear constraints - reflect limitations.
A Linear Program
Max
S.T.
7X1 + 4X2
3X1 + X2 <= 580
2X1 + 5X2 <= 720
X1
>= 20
X2 <= 100
X1, X2 >= 0
Format of a Linear Program
• No variable is in denominator.
• At most one term for each variable.
• Variable terms are at left, constant
terms are at right (called right-handside, RHS).
• Align columns of inequality signs,
variable terms, and constants.
• Put non-negative constraints in at last.
Solution
• A solution is a set of values each for a
variable.
• A feasible solution satisfies all
constraints.
• An infeasible solution violates at least
one constraint.
• The optimal solution is a feasible
solution that meets the objective.
LP Solution Methods
• Trial-and-Error
(brute force)
• Graphic Method
(Won’t work if more than 2 variables)
• Simplex Method
(Elegant, but time-taking if by hand)
• Computerized simplex method
(We’ll use it programmed in QM)
Process of Solving a Problem By
Using LP
Step 1. Formulate the problem into a linear
program (by us)
Step 2. Solving the linear program (by
computer)
Step 3. Understanding the result and
sensitivity analysis (by us / computer)
LP Formulation
• Before using QM to solve a problem, we
must first formulate the problem into a
linear program, which is a description of
the problem in terms of LP.
• Therefore, the process of formulating a
problem in LP is a process of describing
the problem by using an objective function
and a couple of constraints.
LP Example 1, p.32-33
Resource Requirements
Labor
Clay
Products
(hr/unit)
(lb/unit)
Bowl
1
4
Mug
2
3
Resource
40 hours
120 lb
Available
Profit
($/unit)
40
50
Find how many bowls and mugs should be produced to
maximize the profit.
Steps for LP Formulating
• Define variables unambiguously.
• Describe the objective function by using
the variables.
• Describing restrictions one at a time by
using the variables, which form
constraints.
LP Example 2 p.47-49
Brand
Super-gro
Crop-quick
Minimum
requirements
Chemical Contribution
Nitrogen Phosphate
(lb/bag)
(lb/bag)
2
4
4
3
16
Cost
($/bag)
6
3
24
How many bags of each brand should be
purchased in order to minimize the total cost?
Irregular LP Problems
• A regular LP has one optimal solution.
• Irregular cases:
– Multiple optimal solutions
– Infeasible problem
– Unbounded problem
Chapter 3
Linear Programming:
Sensitivity Analysis
Sensitivity Analysis (SA)
• SA is the analysis of the effect of
parameter changes on the optimal
solution.
• SA is conducted after the optimal
solution is obtained.
Shadow Price (Dual Value)
• A shadow price is associated with a
constraint in the solution.
In a product-mix problem
• as in example of ‘bowls and mugs’, a shadow
price means:
– the marginal value of a resource, i.e.,
– the contribution of an additional unit of a
resource to the objective function value,
i.e.,
– The highest “price” the company would be
willing to pay for one additional unit of a
resource.
What Is “Dual”?
• Each linear program has another LP
associated with it. They are called a pair
of primal and dual.
• The dual LP is the “transposition” of the
primal LP.
• Primal and dual have equal optimal
objective function values.
• The solution of the dual is the shadow
prices of the primal, and vice versa.
More General on Shadow Price:
• The shadow price of a constraint shows how
much the objective function value would be
better off if there were one unit increase on
the RHS of the constraint.
• A shadow price can be negative, which
shows a negative contribution (i.e., worse
off) to the objective function value by an
additional unit of RHS of the constraint.
S.A. on RHS
• Sensitivity range for a RHS value is the
range over which the RHS value can
change without changing the current
shadow price.
• Sensitivity range for a RHS value is
also the range over which the RHS
value can change without changing the
non-zero variable mix in the solution.
S.A. on Objective Coefficients
• Sensitivity range for an objective
coefficient is the range over which the
objective coefficient can change
without changing the current optimal
solution.
S.A. on other changes
• To see sensitivities on following
changes, one must solve the changed
LP again:
– Changing constraint coefficients
– Adding a new constraint
– Adding a new variable
Why doing S.A.?
• LP is used for decision making on
something in the future.
• Rarely does a manager know all of the
parameters exactly. Many parameters
are inaccurate “estimates” when a
model is formed and solved.
• We want to see to what extent the
optimal solution is stable to the
inaccurate parameters.
Sensitive or In-sensitive?
• Do we want a model more sensitive or
less sensitive to the inaccuracies
(changes) of parameters in it ?
• Answer:
Less sensitive.
• Why?
– An optimal solution that is insensitive to
inaccuracies of parameters is more likely
valid in the real world situation.
Chapter 4
Linear Programming:
Modeling Examples
LP Modeling
• To model a decision making problem
with LP:
– Understand the problem thoroughly;
– Identify the variables and objective;
– Describe the problem in terms of the
variables, objective function, and
constraints.
Examples Covered:
•
•
•
•
•
Product mix
Investment
Marketing
Blend (?)
Transportation (?)
A product Mix Example. p.112
Work hour
Space taken
Cost($)/dozen
Profit($)/dozen
no more than
Sweatshirts
T-shirts
Printing on
Printing on Printing on Printing on
front side
both sides
front side
both sides
0.1/dozen
0.25/dozen
0.08/dozen 0.21/dozen
3 std.
3 std.
1 std.
1std.
boxes/dozen boxes/dozen box/dozen box/dozen
36
48
25
35
90
125
45
65
500 dozen
500 dozen
Capacity
available
72 hours
1200 std.
boxes
$25,000
How many of each of four products should be produced so that the total profit is
maximized ?
An Investment Example. p.120-122
Annual Return
municipal
bonds
8.50%
Investment Alternatives
treasury
growth stock
CDs
bills
fund
5.00%
6.50%
13.00%
* Municipal bonds <= 20%.
* CDs <= sum of other three.
* (Treasury bills + CDs) >= 30%.
* (Treasury bills + CDs) / (municipal bonds + stock fund) >= 1.2/1.
* Total amount of investment is up to $70,000.
How much should be put in each of the four investment alternatives
so that the total return is maximized ?
A Marketing Example. p.126-127
Types of Advertising
television
radio
newspaper
commercials
commercials
ads
Exposures
(people / ad)
Cost $
limits on number
of ads
limits on total
number of ads
Total budget
limit
20,000
12,000
9,000
15,000
6,000
4,000
<=4
<=10
<=7
<=15
$100,000
How many ads of each type should be used so that the total
exposure is maximized ?
A Blend Example, p.133-135
How mnay barrels of each grade of motor oil (made of three componets) should be
produced to maximize total profit?
Component 1 Component 2 Component 3 Selling Price
Minimum
Super Grade
>=50%
<=30%
----
$23
3,000 barrels
Premium
Grade
>=40%
----
<=25%
$20
3,000 barrels
Extra Grade
>=60%
>=10%
----
$18
3,000 barrels
Barrels
Available
4,500
2,700
3,500
Cost $/barrel
$12
$10
$14